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The of Organic Compounds in Supercritical CO2 Guillermo A. Alvareza, Wolfram Baumannb, Martha Bohrer Adaimeb, and Frank Neitzelb a Departamento de Qu´ımica, Facultad de Ciencias, Universidad de los Andes, Carrera 1 No. 18A-10/70, Bogot´a, Colombia b Institute of Physical Chemistry, University of Mainz, D-55099 Mainz, Germany Reprint requests to Prof. G. A. A.; Fax: 00571 3324366; E-mail: [email protected]

Z. Naturforsch. 60a, 641 – 648 (2005); received May 3, 2005

A simple solution model is proposed to describe the effect of solvent-solute interactions on the solubility of nonpolar and slightly polar substances in supercritical solvents. Treating the system as an , the effect of pressure on the solubility is zero or nearly zero, as it is governed by the difference in molar volume of the pure supercooled liquid solute and the pure solute, and this may be nearly zero. Deviations from ideal behavior are given by activity coefficients of the Margules type with the interaction parameter w interpreted as interchange as in the lattice theory. The hypothesis is put forward that the interchange energy is of the same form as a function proposed by Liptay and others to describe the effect of the solvent on the wavelength of the absorption maximum of the solute dissolved in the solvent. The function consists of a radius of interaction a and a function g(ε) of the dielectric constant ε of the solvent, treated as a continuum. The function g depends on pressure through the pressure dependence of the dielectric constant ε(P). The attractive feature of this formalism, introduced by Baumann et al. and here justified thermodynamically, is that plots of the logarithm of solubility vs. g are linear, except for polar solutes near the solvent’s critical point. Changes in slope then admit interpretation as changes in the radius of interaction a with possible clues about the mechanism of of these .

Key words: Dielectric Interactions; Solubility; Supercritical CO2; Thermodynamic Model; Activity Coefficient.

1. Introduction solvent depends nearly linearly on the density [3], one may substitute the dielectric constant ε(P) by the den- The solubility of substances in supercritical sol- sity ρ(P) as a measure of solvent-solute interactions in vents is relevant in separation processes such as su- practical cases. percritical fluid extraction and chromatography and However, we can proceed further according to the for modeling the state of a solute in a sol- theories of Lippert [4], Mataga et al. [5] and no- vent [1]. In particular, since near the critical point the tably Liptay [6], who showed that in polar solvents solvent’s density can change by a factor of ten with the change due to the solvent in the wavelength at only small changes in the and the pres- the maximum of absorption of light by the solute sure, it is clear that the solubility of the solute can vary or solvatochromic shift is a function of (ε(P) − 1)/ likewise by a large factor. Enormous changes in the (a3(2ε(P)+1)) = g(ε)/a3. Further, the shift in so- nature of the solvent, quantified for example by the in- lution is towards red relative to the , that terchange energy w of the lattice theory, would be re- is the solvent-solute system is stabilized by disper- quired to bring about the simple “distance” effect due sion interactions and by interactions of the dipole of to the compressibility of the solvent near the critical the molecule with the surrounding dielectric. Since the point. function ε(P) increases monotonically with P, this sta- It thus appears that the solvent’s density for a solvent bilizing effect for constant a always increases the sol- near the critical point is a suitable measure of solvent- ubility, reaching an asymptotic level at high P (but see solute interaction and hence of solubility. Accordingly, below). Thus, here the interchange energy is negative, plots of the logarithm of solubility vs. the density or leading to negative deviations of Raoult’s law, in con- the logarithm of the density obey a linear relationship trast to the usual positive deviations for nonpolar so- in many cases [2]. Since the dielectric constant of the lutes in nonpolar solvents.

0932–0784 / 05 / 0800–0641 $ 06.00 c 2005 Verlag der Zeitschrift f¨ur Naturforschung, T¨ubingen · http://znaturforsch.com 642 G. A. Alvarez et al. · Solubility in Supercritical CO2 Numerous models have been proposed for the ef- 2. Thermodynamic Model fect of the thermodynamic variables temperature T It is often the case for solutes with high and pressure P on the fraction xB of solute B in the saturated solution since the work of Schr¨oder points that solid B dissolves in liquid A at a given pres- on the effect of temperature on the solubility of naph- sure only up to a certain xB called the thalene in (see e.g. Pitzer [7]). The present equilibrium mole fraction or solubility of the solute in formalism follows the same steps of Schr¨oder but the solution saturated with the solid. For component B, considering changes of pressure instead of changes equilibrium between a solid phase s and a liquid phase of temperature. Whereas molar is the vari- l is dictated by the equality of chemical potentials able that controls ideal solubility as a function of µs = µl . temperature, here molar volume controls ideal solu- B B (1) bility as a function of pressure. In contrast to the The of the pure solid phase B is temperature effect, the ideal pressure effect is, how- ever, very small or zero, the changes in solubility µs = µ0s, B B (2) arising from real interactions between solvent and solute. where the upper index 0 denotes a pure phase. For the The model of Kurnik and Reid [8] is possibly one component B in the mixed phase, of the most comprehensive models of solubility of µl = µ∗l( , )+ . or in compressed . Using volumet- B B T P RT lnaB (3) ric data at constant temperature and composition and µ∗l( , ) some mixing rules, they predict ideal solubility ini- B T P is the chemical potential of the supercooled tially decreasing with pressure, passing through a min- pure liquid B with the dependencies on temperature imum, rising dramatically again as the phase and pressure indicated explicitly. aB is the activity of goes through the critical point and, apparently, going component B in solution and R is the universal gas con- through a maximum at very high pressure and decreas- stant. ing solubility thereafter due to “repulsive forces” be- Replacing (2) and (3) in (1) gives: tween solute and solvent. The steep rise in solubility µ0s = µ∗l + RT a . near the critical point is attributed to real attractive in- B B ln B (4) teractions as evidenced by a vapor-phase fugacity coef- Consider the effect of pressure on activity at T = ficient in the high-pressure gas mixture, which is much constant. Taking derivatives with respect to pressure less than one. The present model, starting from an ideal       liquid solution and describing deviations from ideal be- ∂µ0s ∂µ∗l ∂(lna ) B = B + RT B . (5) havior through an energy parameter suggested by the ∂P ∂P ∂P work of Liptay [6] may be considered a simplification T T T of the model of Kurnik and Reid [8]; both models at- For a pure substance tribute enhanced solubility near the critical point to real     attractive interactions between solute and solvent. The ∂µ0 ∂µ0 µ0 = B + B , present model cannot predict the minimum in solubil- d B ∂ dP ∂ dT (6) P T T P ity reported by Kurnik and Reid [8] (which may be of commercial importance, for example if a stream of gas where   is to be kept as free as possible of impurities; but see ∂µ discussion below) and as mentioned above, neither can = ∂ Vm (7) the maximum reported by them be predicted, as the P T type of forces responsible for the solvatochromic shift and are always attractive. A refined model, which takes into   account association of polar solutes for large values of ∂µ = − . the interaction parameter and/or large solute concen- ∂ Sm (8) T P tration, is explained below the discussion. This effect may contribute or eventually account for the mentioned Vm and Sm are the molar volume and the molar entropy, maximum. respectively, of the pure substance. For a process at G. A. Alvarez et al. · Solubility in Supercritical CO2 643 constant temperature the second term on the right in (6) T = 300 K. This leaves ideal solubility practically in- vanishes. Substituting (7) in (5) yields: dependent of pressure up to say 30 MPa.   For a real solution a = γ x , where γ is the activ- ∂( ) B B B B 0s = ∗l + lnaB ity coefficient of the solute B in the solution. Replacing Vm Vm RT ∂ (9) P T this in (13) and neglecting the ∆fusionVm term gives or rearranging lnxB = c0 − lnγB. (15)   ∂( ) 0s − ∗l ∆ lnaB = Vm Vm = − fusionVm , For the activity coefficient we adopt the Margules ∂ (10) P T RT RT equation where ∆ V is the change in molar volume on melt- w fusion m lnγ = x2 . (16) ing the solid to the supercooled liquid. Presumably, B kT A ∆fusionVm is a positive quantity, so that the slope in (10) is negative and the activity of the solute decreases with xA is the mole fraction of solvent A and k is Boltz- an increase in pressure at constant temperature. Taking mann’s constant. In the original Margules equation w the liquid and solid substances as incompressible over is an energy parameter; in the lattice theory w is the interchange energy given by a limited pressure range, the quantity ∆Vm is a constant, and a simple integration gives   1     w = z ΓAB − (ΓAA +ΓBB) , (17) lna P B ∆fusionVm 2 d(lnaB)=− dP, lna RT P (11) B where Γ is the potential energy of an (A-B) pair and T = const. AB likewise ΓAA and ΓBB of (A-A) and (B-B) pairs, re- This equation is entirely analogous to the Schr¨oder spectively. z is the coordination number of the lattice. equation for solubility as a function of temperature [7]. Physically, w is the energy absorbed in the process of Using P∗ as the pressure of the pure supercooled liquid separating z pairs of type (A-A) and z pairs of type at T we obtain (B-B) to form 2z dissimilar (A-B) pairs. Calculations of w from molecular properties are due to Kohler us- ∆ V lna = − fusion m (P − P∗), (12) ing London’s theory of dispersion forces. Inadequate B RT knowledge of intermolecular forces hampers this type of calculation [1]. and this may be written Liptay [6], and also Lippert [5] and Mataga et al. [6], ∆ V obtained an expression for the effect of the solvent lna = c − fusion m P, (13) B 0 RT on the wavenumber of absorption of the dissolved molecule. The main part of this calculation is the es- where c0 is a constant. timation of the energy of the dissolved molecule rel- ative to its energy in the gas phase; while the change 3. The Activity Coefficient in energy is attributed to the dissolved molecule, nec- essarily both solute and surrounding solvent molecules ≈ For an ideal solution aB xB, the mole fraction of are involved in this change. For the estimation of the B in solution. Then (13) becomes energy, the model of Onsager [9] is used, according to which the molecule is located at the center of a ∆fusionVm lnxB = c0 − P. (14) cavity within an isotropic and homogeneous dielectric RT medium, which represents the solvent. Under the as- With ∆fusionVm > 0, this equation is reminiscent of sumption that the permanent dipole of the molecule in µ 0 Kurnik and Reid’s result [8] that solubility in the ideal the ground state g is parallel to the permanent dipole gas decreases with an increase in pressure. On the of the molecule in the given (Franck-Condon) excited µ0 other hand, for a typical solid or liquid molar volume state a and neglecting polarizability terms, the final −4 3 of 10 m and a volume change of melting of 10% result for the wavenumber ν˜a of the absorption maxi- −9 −1 of the volume, the factor ∆Vm/RT ≈ 4 · 10 Pa at mum of the molecule dissolved in an unpolar solvent 644 G. A. Alvarez et al. · Solubility in Supercritical CO2 is [10]   1 0 2 1 0 2 hcν˜ = const.− f hcD + (µ  ) − (µ ) . (18) a 2 a 2 g h is Planck’s constant, c the speed of light, and the term − f hcD represents the difference in dispersion interac- tions between ground and excited states. D is neglected for polar molecules. f is a positive quantity required in the description of the cavity. For a spherical cavity of radius a and the dipole moment of the molecule repre- sented by a point dipole at the center of the cavity, one has ε − = 1 2 1 = 1 , f 3 3 g (19) 4πε0 a 2ε + 1 2πε0a where ε0 is the permittivity of vacuum. The negative sign in front of f in (18) to the Fig. 1. g-P isotherms for CO2 from [3] showing liquid (L), vapor (V), coexisting liquid and vapor (L+V) and supercriti- red shift toward lower energy from the gas to the dis- = . = . µ 0 ≥ µ0 cal fluid (SCF) regions. PC 7 38 MPa, TC 304 1K. solved molecule for the usual case, where a g . The validity of (18) was tested meticulously by Lip- B. The data in [10] are given in units of mmol of B per ν tay [6], who plotted ˜a against g for many solvents of Liter, c , and were converted to mole fraction of B, x , ε B B known and obtained excellent straight lines. in the following way [10]: Noting that hcν˜a represents the energy of interaction of the solute with the solvent, Rodrigues et al. [11] and nB = nB ≈ nB = V = cB , Neitzel [10] began to plot log solubility vs. g and ob- xB nA (22) nB + nA nA ρA tained very good straight lines in many cases. Formally V then, we adopt the following form for the exchange en- nB is the number of moles of B, nA is the number of ergy based on (18): moles of A and ρ A is the molar density of solvent A in mmol/l at the given pressure. The cB data in [10] were c2 w = c1 − g, (20) measured spectroscopically from a reference concen- a3 tration using Lambert-Beer’s law. and this may be substituted in (15) with (16), redefin- Two solvents, dioxide, which is an unpolar ing c1 and c2 for constant T and xB → 0: molecule, and trifluoromethane, which is a slightly po- lar solvent were investigated in [10]. ε-ρ-P isotherms = + c1 . ε lnxB c0 3 g (21) for CO2 were taken from [3]; for CF3H, -P isotherms a over the whole range and ε-ρ isotherms from 0 to − This is the equation used in [10] to correlate a relatively 100 kg m 3 were found in [12]. A long extrapolation − extensive series of experimental results. If needed, the up to 700 kg m 3 was then made of the latter data with temperature which appears in (16) may be passed ex- the help of the Clausius-Mossotti equation. This was plicitly onto this equation to correlate data at dif- necessary to obtain the ρ-P isotherm for the conversion ferent . A brief summary of the results of the data in [10] to mole fraction according to (22); it follows. was verified that the result of the extrapolation within reasonable limits has negligible effect on the results 4. Results and Discussion shown below (see Fig. 5). It is instructive to consider first the g-P isotherms for In order to demonstrate the applicability of the pro- CO2 (Fig. 1) as the underlying cause of the phenomena posed model, data from [10] and [2] are used which of solubility investigated here. Over the pressure range have carefully been read from the published graphs. from 5 to 16 MPa at T = 313.15 K, where most of

The data in [2] are taken directly as mole fraction of the data lie, gCO2 increases almost tenfold. As the fluid G. A. Alvarez et al. · Solubility in Supercritical CO2 645

10+ln xB 1

0.5

0

-0.5

-1

-1.5

Dienochlor -2 Isoproturon

-2.5 51015 20 25 P/MPa Fig. 2. Ln solubility vs. g for the nonpolar pesticides iso- Fig. 3. Ln solubility vs. pressure of the isoproturon and proturon and dienochlor in superfluid carbon dioxide at T = . dienochlor data in Fig. 2, with the regression lines of Fig. 2 313 15 K. replotted in this graph. approaches incompressibility the isotherms of g vs. P level off. The similarity with ρ-P isotherms is appar- ent but the increase in ρ is only about eightfold at this temperature. The simplest data to analyze are nonpolar less volatile solutes such as the pesticides isoproturon and dienochlor in the nonpolar solvent carbon dioxide. In Fig. 2 the lnxB vs. g experimental data points are very well approximated by straight regression lines with correlation coefficients r = 0.993 and 0.987, respec- tively. To demonstrate the correctness of the procedure em- ployed in Fig. 2, the same solubility data are replot- ted in Fig. 3 against pressure along with the solubil- ity predicted by the linear regression of Figure 2. In a remarkable way, the regression line of Fig. 2 repro- duces the complex S-shaped experimental curve prac- tically exactly. Note that, plotted in this fashion, the present S-shaped curve bears resemblance to the sim- ilarly S-shaped curve of Kurnik and Reid [8]. In the Fig. 4. Ln solubility vs. g for the slightly polar same spirit, one may extrapolate the regression line to- donor-acceptor molecule 5-dimethylamino-5’-nitro-2,2’-bi- = . ward zero pressure until it meets the ideal gas law [1] thiophene in superfluid carbon dioxide at T 313 15 K showing normal linear behavior in this plot. s PB xB = , (23) of the minimum in solubility mentioned in the intro- P duction. s where PB is the saturation (vapor) pressure of pure Similar results are obtained for polar nonvolatile solid B and P is the pressure. This would give an idea molecules such as the donor-acceptor conjugated 5- 646 G. A. Alvarez et al. · Solubility in Supercritical CO2 made that this slope contains molecular quantities such as dipole moments and polarizabilities, which describe the solubility. To improve the solubility of the polar molecule 5- dimethylamino-5’-nitro-2,2’-bithiophenea slightly po- lar solvent, trifluoromethane, was used. The plot of ln solubility vs. g at T = 313.15 K is no longer linear as shown in Fig. 5, assuming the ε-ρ-P data are correct. Similarly, the respective plot of lnxB versus g for more volatile compounds is not linear, as is shown in Fig. 6 for three polyaromatic hydrocarbons, naph- thalene at 328.15 K from Spiliotis et al. [13] and for phenanthrene and phenylanthracene at 313.15 K from [10].

5. A More Refined Theory

A modified theory known as the quasichemical ap- Fig. 5. Ln solubility vs. g for the donor-acceptor molecule in Fig. 4 dissolved in the slightly polar fluid trifluoromethane proximation was introduced by Guggenheim (see [1]) at T = 313.15 K showing some deviation from the linear re- to account for ordering effects for example when polar lation proposed here assuming the data for molecules are dissolved in nonpolar solvents without the solvent are correct. change of volume. The polar solute molecules will tend to surround themselves with other solute molecules in dimethylamino-5’-nitro-2,2’-bithiophene in the non- polar supercritical carbon dioxide solvent, as shown in favor of the solvent-solute contacts thus lowering solu- Figure 4. A good straight line is obtained again, the bility. This may explain the downturn in the straight g increased scatter being attributed to experimental de- line observed at high for the polar solutes e.g. 5- dimethylamino-5’-nitro-2,2’-bithiophene in Figure 5. tection difficulties due to the lower solubility of the K polar solute in the nonpolar solvent. The slope of this An equilibrium constant is defined for the “reac- plot is 86.4 in contrast to the lesser slopes of 26.4 tion” and 27.1 reported for the nonpolar solutes isoproturon (A-A)+(B-B)=2(A-B). (24) and dienochlor in Fig. 2 in accord with the assumption

Fig. 6. Ln solubility vs. g for some pol- yaromatic hydrocarbons in CO2 at T = 313.15 K. The regression curve is cal- culated according to the cubic equa- tion (33). G. A. Alvarez et al. · Solubility in Supercritical CO2 647 ∗ Setting where uE is the excess energy of the random mixture

2w E∗ ∆ u = N (25) u = NAwxAxB (27) r A z β for the molar energy change of the “reaction” with N A and the parameter is: Avogadro’s number and this in the van’t Hoff equa-    1/2 2w/zkT tion, integrating at constant volume and using some β = 1 + 4 e − 1 xAxB . (28) conservation relations gives for the excess energy of E mixing u The excess Helmholtz and excess Gibbs are   obtained by integrating the Gibbs-Helmholtz equation E E∗ 2 u = u , (26) at constant volume (vE = 0) with the result [1]: β + 1

  E E a g z β − 1 + 2xA β − 1 + 2xB = = xA ln + xB ln . (29) RT RT 2 xA(β + 1) xB(β + 1)

For small (w/zkT ), expansion of the exponential, logarithmic and binomial terms up to quadratic terms in both w and xAxB yields:         gE w 1 2w 1 2w 2 1 2w 2 2w 2 = x x 1 − x x − + x x − (x x )2+O3(w,x x ) . (30) RT kT A B 2 zkT A B 2 zkT 2 zkT A B zkT A B A B

The term in square brackets shows contributions to the excess Gibbs energy up to second order both due to large w alone and to large w and xAxB combined. That is, even at infinite dilution (xB → 0) there can be ordering effects, e.g. of solute pairs if w is large. From thermodynamics,

E ∂ nTg RT lnγB = ∂nB T,P,nA         (31) w 1 2w 2 1 2w 1 2w 2 = x2 1 − + x (1 − 3x ) − x [1 − 3x + 2x x (2 − 3x )] ; kT A 2 zkT 2 zkT B A 2 zkT B A A B A

nT is the total number of moles, nT = nA + nB. At infi- high negative values of w (high g). This is surprisingly nite dilution xA → 1, xB → 0, the final result is: what the data in Figs. 5 and 6 show. The correction   depends on z, the coordination number of the lattice. 2 w 1 2w If w from (20) is put into (33) the solubility data lnx lnγ = 1 − . (32) B B kT 2 zkT should be able to be represented by a respective cubic plot against g. Indeed, Fig. 6 shows that the data points Replacing this in (15) gives: can very well be simulated by a model curve, cubic   in g. w 2 w 3 lnx = c − + . (33) Several other theories are known to deal with B 0 kT z2 kT nonideality of polar solutes in nonpolar solvents. Since w is negative, this equation shows that the ran- These are the Wilson equation, the NRTL (nonran- dom theory predicts an increase in solubility over the dom, two-liquid) equation and the UNIQUAC (uni- (nearly constant) ideal term c0 but, since the last term versal, quasichemical) theory for molecules of dif- on the right is cubic, the quasichemical approximation ferent sizes. Details of these theories are found provides a correction which diminishes the solubility at in [1]. 648 G. A. Alvarez et al. · Solubility in Supercritical CO2 Conclusion ing. The solutions are then extremely nonideal in the sense of Raoult’s law. A model is proposed to account for the effect of The solvent-solute interaction is modeled according pressure on the solubility of nonpolar and slightly po- to the theory of Liptay for the effect of solvent on dis- lar solutes in supercritical solvents. The development placement of the absorption maximum of the molecule follows the steps of Schr¨oder, see e. g. [7], but taking from the gas state to the solvated state. into account changes of pressure instead of changes of temperature. In contrast to the ideal temperature effect, Acknowledgements which is governed by the enthalpy of fusion, the ideal G. A. A. wishes to thank the Deutscher Akademi- pressure effect is controlled by the change of volume scher Austauschdienst (DAAD) for a generous schol- on fusion, and this is a very small quantity. arship to stay at the University of Mainz, Germany According to the present model, the much enhanced during the summer of 2002. For providing us with solubility observed for solutes near and beyond the samples of bithiophene we are indebted to Prof. R¨udi- critical point is attributed to the attractive interactions ger Wortmann (†) from the Universit¨at Kaiserslautern, between solute and solvent, which are rapidly increas- Germany.

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